Integral echelon basis of level-one modular forms

If $N+1=\dim M_k(S{L}_2(\mathbb{Z}))$, there is a basis $f_0,\dots ,f_N$ with integral Fourier coefficients andStart from the integral forms ${\Delta }^i{E}_4^a{E}_6^b$ and use integer elimination on their unitriangular leading coefficients.

If the first nonconstant coefficient of $E_k$ has reduced denominator divisible by a prime $p$, the integral echelon basis produces an integral cusp form whose Fourier coefficients are congruent modulo $p$ to ${\sigma }_{k-1}(n)$.

The weight-sixteen cusp form space is spanned by $E_4\Delta$. Its leading coefficient and the Fourier expansion of a normalized Eisenstein series give $3617E_{16}=3617E_4^4-3456000E_4\Delta$. The coefficient multiplier $-3456000$ is congruent to $16320$ modulo $3617$, and is invertible there. Coefficient comparison gives the displayed modular congruence. Integrality of $E_4\Delta$ follows from the modular discriminant product.